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Abstract: A bivariate rational interpolation surface based on function values has been constructed in the authors’ earlier works. This paper deals with the convexity control of interpolating surfaces. The sufficient and necessary conditions for interpolating surfaces to be convex are derived. The convexity of the interpolating surface can be changed locally by selecting suitable parameters under the condition that the interpolation data are not changed. Examples are given to show how the parameters can be chosen and the shapes of the surfaces changed. [Copyright &y& Elsevier]

Abstract: In this paper a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters, and it can be expressed by the symmetric bases. It is proved that the interpolation is stable. The concept of integral weights coefficients of the interpolation is given, which describes the “weight” of the interpolation points and the quantity as the interpolation data in the local interpolating region. [Copyright &y& Elsevier]

Abstract: In this paper a bivariate rational Hermite interpolation is developed to create a space surface using both function values and the first-order partial derivatives of the function being interpolated as the interpolation data. The interpolation function has a simple and explicit rational mathematical representation with parameters. There are two schemes to construct the interpolation functions, an example shows that both of them approximate the function being interpolated very well. The basis of this interpolation is derived and the interpolating function can be C 1 in all the interpolation region when the parameters satisfy a simple condition. When a patch of the interpolating surface does not satisfy the need of design, say it is too high or too low at a point and its neighbourhood, the values of the interpolating function in these points need to be decreased or increased under the condition that the interpolating data are not changed. For this, sufficient conditions are derived, and it is shown that this can be done just by selecting suitable parameters in the interpolation. [Copyright &y& Elsevier]

Abstract: A bivariate rational interpolation method with parameters was created in an earlier work which was based on function values only. This paper will deal with the bounded property and the point control method of the interpolating surface. It is proved that the values of the interpolating function in the interpolation region are bounded no matter what the parameters might be; this is called the bounded property of the interpolation. Also, the approximation expressions of the interpolation are derived; they do not depend on the parameters. More important is that the value of the interpolating function at any point in the interpolating region can be modified under the condition that the interpolating data are not changed by selecting suitable parameters, so the interpolation surface may be modified for the given interpolation data when needed in practical design. In the special case, the “Central Point-Mean Value” control is studied, and an example is given to show the control. [Copyright &y& Elsevier]